Profiling & Parallelization

Lecture 21

Dr. Colin Rundel

Profiling & Benchmarking

profvis demo

n = 1e6
d = tibble(
  x1 = rt(n, df = 3),
  x2 = rt(n, df = 3),
  x3 = rt(n, df = 3),
  x4 = rt(n, df = 3),
  x5 = rt(n, df = 3),
) |>
  mutate(y = -2*x1 - 1*x2 + 0*x3 + 1*x4 + 2*x5 + rnorm(n))
profvis::profvis(lm(y~., data=d))

Benchmarking - bench

d = tibble(
  x = runif(10000),
  y = runif(10000)
)

(b = bench::mark(
  d[d$x > 0.5, ],
  d[which(d$x > 0.5), ],
  subset(d, x > 0.5),
  filter(d, x > 0.5)
))
# A tibble: 4 × 6
  expression                 min   median `itr/sec` mem_alloc `gc/sec`
  <bch:expr>            <bch:tm> <bch:tm>     <dbl> <bch:byt>    <dbl>
1 d[d$x > 0.5, ]           128µs    140µs     7002.  239.84KB     19.4
2 d[which(d$x > 0.5), ]    137µs    152µs     6388.  271.49KB     33.5
3 subset(d, x > 0.5)       167µs    195µs     4940.  288.85KB     26.3
4 filter(d, x > 0.5)       379µs    432µs     2167.    1.48MB     35.4

Larger n

d = tibble(
  x = runif(1e6),
  y = runif(1e6)
)

(b = bench::mark(
  d[d$x > 0.5, ],
  d[which(d$x > 0.5), ],
  subset(d, x > 0.5),
  filter(d, x > 0.5)
))
# A tibble: 4 × 6
  expression                 min   median `itr/sec` mem_alloc `gc/sec`
  <bch:expr>            <bch:tm> <bch:tm>     <dbl> <bch:byt>    <dbl>
1 d[d$x > 0.5, ]            12ms   12.4ms      80.6    13.4MB     60.4
2 d[which(d$x > 0.5), ]   13.3ms   13.6ms      73.6    24.8MB    210. 
3 subset(d, x > 0.5)      17.7ms   17.9ms      55.7    24.8MB     65.0
4 filter(d, x > 0.5)      13.5ms   13.9ms      71.9    24.8MB     56.8

bench - relative results

summary(b, relative=TRUE)
# A tibble: 4 × 6
  expression              min median `itr/sec` mem_alloc `gc/sec`
  <bch:expr>            <dbl>  <dbl>     <dbl>     <dbl>    <dbl>
1 d[d$x > 0.5, ]         1      1         1.45      1        1.06
2 d[which(d$x > 0.5), ]  1.11   1.09      1.32      1.86     3.70
3 subset(d, x > 0.5)     1.47   1.45      1         1.86     1.14
4 filter(d, x > 0.5)     1.12   1.12      1.29      1.86     1   

t.test

Imagine we have run 1000 experiments (rows), each of which collects data on 50 individuals (columns). The first 25 individuals in each experiment are assigned to group 1 and the rest to group 2.

The goal is to calculate the t-statistic for each experiment comparing group 1 to group 2.

m = 1000
n = 50
X = matrix(
  rnorm(m * n, mean = 10, sd = 3), 
  ncol = m
) |>
  as.data.frame() |>
  set_names(paste0("exp", seq_len(m))) |>
  mutate(
    ind = seq_len(n),
    group = rep(1:2, each = n/2)
  ) |>
  as_tibble() |>
  relocate(ind, group)
X
# A tibble: 50 × 1,002
     ind group  exp1  exp2  exp3  exp4  exp5  exp6
   <int> <int> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
 1     1     1 12.1  15.0  12.8   5.54 12.0  11.5 
 2     2     1  5.61  6.99 14.3   8.47  6.49  5.17
 3     3     1 12.5  11.6  15.7  10.3  14.0   7.52
 4     4     1  8.33  8.39 10.5  15.6   9.94  8.21
 5     5     1 11.1   5.80  8.22  9.48  5.08  2.59
 6     6     1  9.48  6.19  6.05  9.16  8.69 17.0 
 7     7     1 11.2  11.8   5.98  6.05  9.70 12.9 
 8     8     1  9.56  8.14 14.6  11.4   7.39  9.63
 9     9     1 10.1   8.72 10.3   7.71 11.5  12.1 
10    10     1 17.1  11.6   8.38 12.3   7.88  8.59
# ℹ 40 more rows
# ℹ 994 more variables: exp7 <dbl>, exp8 <dbl>,
#   exp9 <dbl>, exp10 <dbl>, exp11 <dbl>,
#   exp12 <dbl>, exp13 <dbl>, exp14 <dbl>,
#   exp15 <dbl>, exp16 <dbl>, exp17 <dbl>,
#   exp18 <dbl>, exp19 <dbl>, exp20 <dbl>,
#   exp21 <dbl>, exp22 <dbl>, exp23 <dbl>, …

Implementations

ttest_formula = function(X, m) {
  for(i in 1:m) t.test(X[[2+i]] ~ X$group)$stat
}

system.time(ttest_formula(X,m))
   user  system elapsed 
  0.183   0.001   0.186 
ttest_for = function(X, m) {
  for(i in 1:m) t.test(X[[2+i]][X$group == 1], X[[2+i]][X$group == 2])$stat
}

system.time(ttest_for(X,m))
   user  system elapsed 
  0.064   0.001   0.066 
ttest_apply = function(X) {
  f = function(x, g) {
    t.test(x[g==1], x[g==2])$stat
  }
  apply(X[,-(1:2)], 2, f, X$group)
}

system.time(ttest_apply(X))
   user  system elapsed 
  0.053   0.000   0.054 

Implementations (cont.)

ttest_hand_calc = function(X) {
  f = function(x, grp) {
    t_stat = function(x) {
      m = mean(x)
      n = length(x)
      var = sum((x - m) ^ 2) / (n - 1)
      
      list(m = m, n = n, var = var)
    }
    
    g1 = t_stat(x[grp == 1])
    g2 = t_stat(x[grp == 2])
    
    se_total = sqrt(g1$var / g1$n + g2$var / g2$n)
    (g1$m - g2$m) / se_total
  }
  
    apply(X[,-(1:2)], 2, f, X$group)
}

system.time(ttest_hand_calc(X))
   user  system elapsed 
  0.014   0.000   0.015 

Comparison

bench::mark(
  ttest_formula(X, m),
  ttest_for(X, m),
  ttest_apply(X),
  ttest_hand_calc(X),
  check=FALSE
)
# A tibble: 4 × 6
  expression               min   median `itr/sec` mem_alloc `gc/sec`
  <bch:expr>          <bch:tm> <bch:tm>     <dbl> <bch:byt>    <dbl>
1 ttest_formula(X, m) 197.05ms 199.89ms      5.01    8.24MB     26.7
2 ttest_for(X, m)      68.75ms   69.1ms     14.3     1.91MB     26.8
3 ttest_apply(X)       58.24ms  62.82ms     16.1     3.49MB     26.8
4 ttest_hand_calc(X)    8.61ms   9.21ms     89.6     3.45MB     25.9

Parallelization

parallel

Part of the base packages in R

  • tools for the forking of R processes (some functions do not work on Windows)

  • Core functions:

    • detectCores

    • pvec

    • mclapply

    • mcparallel & mccollect

detectCores

Surprisingly, detects the number of cores of the current system.

detectCores()
[1] 10

pvec

Parallelization of a vectorized function call

system.time(pvec(1:1e7, sqrt, mc.cores = 1))
   user  system elapsed 
  0.016   0.012   0.028 
system.time(pvec(1:1e7, sqrt, mc.cores = 4))
   user  system elapsed 
  0.168   0.151   0.249 
system.time(pvec(1:1e7, sqrt, mc.cores = 8))
   user  system elapsed 
  0.092   0.164   0.152 
system.time(sqrt(1:1e7))
   user  system elapsed 
  0.055   0.018   0.072 

pvec - bench::system_time

bench::system_time(pvec(1:1e7, sqrt, mc.cores = 1))
process    real 
   25ms  24.8ms 
bench::system_time(pvec(1:1e7, sqrt, mc.cores = 4))
process    real 
  150ms   179ms 
bench::system_time(pvec(1:1e7, sqrt, mc.cores = 8))
process    real 
  191ms   223ms 

bench::system_time(Sys.sleep(.5))
process    real 
   64µs   497ms 
system.time(Sys.sleep(.5))
   user  system elapsed 
  0.000   0.000   0.505 

Cores by size

cores = c(1,4,6,8,10)
order = 6:8
f = function(x,y) {
  system.time(
    pvec(1:(10^y), sqrt, mc.cores = x)
  )[3]
}

res = map(
  cores, 
  function(x) {
     map_dbl(order, f, x = x)
  }
) |> 
  do.call(rbind, args = _)

rownames(res) = paste0(cores," cores")
colnames(res) = paste0("10^",order)

res
          10^6  10^7  10^8
1 cores  0.004 0.032 0.371
4 cores  0.032 0.145 2.006
6 cores  0.024 0.138 1.367
8 cores  0.033 0.127 1.265
10 cores 0.033 0.147 1.548

mclapply

Parallelized version of lapply

system.time(rnorm(1e7))
   user  system elapsed 
  0.262   0.004   0.266 
system.time(unlist(mclapply(1:10, function(x) rnorm(1e6), mc.cores = 2)))
   user  system elapsed 
  0.309   0.095   0.268 
system.time(unlist(mclapply(1:10, function(x) rnorm(1e6), mc.cores = 4)))
   user  system elapsed 
  0.322   0.089   0.161 
system.time(unlist(mclapply(1:10, function(x) rnorm(1e6), mc.cores = 8)))
   user  system elapsed 
  0.336   0.145   0.172 
system.time(unlist(mclapply(1:10, function(x) rnorm(1e6), mc.cores = 10)))
   user  system elapsed 
  0.360   0.150   0.179 

mcparallel

Asynchronously evaluation of an R expression in a separate process

m = mcparallel(rnorm(1e6))
n = mcparallel(rbeta(1e6,1,1))
o = mcparallel(rgamma(1e6,1,1))
str(m)
List of 2
 $ pid: int 14040
 $ fd : int [1:2] 4 7
 - attr(*, "class")= chr [1:3] "parallelJob" "childProcess" "process"
str(n)
List of 2
 $ pid: int 14041
 $ fd : int [1:2] 5 9
 - attr(*, "class")= chr [1:3] "parallelJob" "childProcess" "process"

mccollect

Checks mcparallel objects for completion

str(mccollect(list(m,n,o)))
List of 3
 $ 14040: num [1:1000000] -0.7172 -1.8334 -0.0983 0.953 0.1629 ...
 $ 14041: num [1:1000000] 0.235 0.554 0.301 0.977 0.644 ...
 $ 14042: num [1:1000000] 0.448 0.192 0.342 1.386 0.397 ...

mccollect - waiting

p = mcparallel(mean(rnorm(1e5)))
mccollect(p, wait = FALSE, 10)
$`14043`
[1] -0.001305267
mccollect(p, wait = FALSE)
NULL
mccollect(p, wait = FALSE)
NULL

doMC & foreach

doMC & foreach

Packages by Revolution Analytics that provides the foreach function which is a parallelizable for loop (and then some).

  • Core functions:

    • registerDoMC

    • foreach, %dopar%, %do%

registerDoMC

Primarily used to set the number of cores used by foreach, by default uses options("cores") or half the number of cores found by detectCores from the parallel package.

options("cores")
$cores
NULL
detectCores()
[1] 10
getDoParWorkers()
[1] 1
registerDoMC(4)
getDoParWorkers()
[1] 4

foreach

A slightly more powerful version of base for loops (think for with an lapply flavor). Combined with %do% or %dopar% for single or multicore execution.

for(i in 1:10) {
  sqrt(i)
}

foreach(i = 1:5) %do% {
  sqrt(i)   
}
[[1]]
[1] 1

[[2]]
[1] 1.414214

[[3]]
[1] 1.732051

[[4]]
[1] 2

[[5]]
[1] 2.236068

foreach - iterators

foreach can iterate across more than one value, but it doesn’t do length coercion

foreach(i = 1:5, j = 1:5) %do% {
  sqrt(i^2+j^2)   
}
[[1]]
[1] 1.414214

[[2]]
[1] 2.828427

[[3]]
[1] 4.242641

[[4]]
[1] 5.656854

[[5]]
[1] 7.071068
foreach(i = 1:5, j = 1:2) %do% {
  sqrt(i^2+j^2)   
}
[[1]]
[1] 1.414214

[[2]]
[1] 2.828427

foreach - combining results

foreach(i = 1:5, .combine='c') %do% {
  sqrt(i)
}
[1] 1.000000 1.414214 1.732051 2.000000 2.236068
foreach(i = 1:5, .combine='cbind') %do% {
  sqrt(i)
}
     result.1 result.2 result.3 result.4 result.5
[1,]        1 1.414214 1.732051        2 2.236068
foreach(i = 1:5, .combine='+') %do% {
  sqrt(i)
}
[1] 8.382332

foreach - parallelization

Swapping out %do% for %dopar% will use the parallel backend.

registerDoMC(4)
system.time(foreach(i = 1:10) %dopar% mean(rnorm(1e6)))
   user  system elapsed 
  0.298   0.028   0.109 
registerDoMC(8)
system.time(foreach(i = 1:10) %dopar% mean(rnorm(1e6)))
   user  system elapsed 
  0.302   0.032   0.076 
registerDoMC(10)
system.time(foreach(i = 1:10) %dopar% mean(rnorm(1e6)))
   user  system elapsed 
  0.325   0.042   0.069 

furrr / future

system.time( purrr::map(c(1,1,1), Sys.sleep) )
   user  system elapsed 
  0.000   0.000   3.012 
system.time( furrr::future_map(c(1,1,1), Sys.sleep) )
   user  system elapsed 
  0.053   0.008   3.097 
future::plan(future::multisession) # See also future::multicore
system.time( furrr::future_map(c(1,1,1), Sys.sleep) )
   user  system elapsed 
  0.206   0.007   1.451 

Example - Bootstraping

Bootstrapping is a resampling scheme where the original data is repeatedly reconstructed by taking a samples of size n (with replacement) from the original data, and using that to repeat an analysis procedure of interest. Below is an example of fitting a local regression (loess) to some synthetic data, we will construct a bootstrap prediction interval for this model.

set.seed(3212016)
d = data.frame(x = 1:120) |>
    mutate(y = sin(2*pi*x/120) + runif(length(x),-1,1))

l = loess(y ~ x, data=d)
p = predict(l, se=TRUE)

d = d |> mutate(
  pred_y = p$fit,
  pred_y_se = p$se.fit
)

ggplot(d, aes(x,y)) +
  geom_point(color="gray50") +
  geom_ribbon(
    aes(ymin = pred_y - 1.96 * pred_y_se, 
        ymax = pred_y + 1.96 * pred_y_se), 
    fill="red", alpha=0.25
  ) +
  geom_line(aes(y=pred_y)) +
  theme_bw()

Bootstraping Demo

What to use when?

Optimal use of parallelization / multiple cores is hard, there isn’t one best solution

  • Don’t underestimate the overhead cost

  • Experimentation is key

  • Measure it or it didn’t happen

  • Be aware of the trade off between developer time and run time

BLAS and LAPACK

Statistics and Linear Algebra

An awful lot of statistics is at its core linear algebra.


For example:

  • Linear regession models, find

\[ \hat{\beta} = (X^T X)^{-1} X^Ty \]

  • Principle component analysis

    • Find \(T = XW\) where \(W\) is a matrix whose columns are the eigenvectors of \(X^TX\).

    • Often solved via SVD - Let \(X = U\Sigma W^T\) then \(T = U\Sigma\).

Numerical Linear Algebra

Not unique to Statistics, these are the type of problems that come up across all areas of numerical computing.

  • Numerical linear algebra \(\ne\) mathematical linear algebra

  • Efficiency and stability of numerical algorithms matter

    • Designing and implementing these algorithms is hard
  • Don’t reinvent the wheel - common core linear algebra tools (well defined API)

BLAS and LAPACK

Low level algorithms for common linear algebra operations

BLAS

  • Basic Linear Algebra Subprograms

  • Copying, scaling, multiplying vectors and matrices

  • Origins go back to 1979, written in Fortran

LAPACK

  • Linear Algebra Package

  • Higher level functionality building on BLAS.

  • Linear solvers, eigenvalues, and matrix decompositions

  • Origins go back to 1992, mostly Fortran (expanded on LINPACK, EISPACK)

Modern variants?

Most default BLAS and LAPACK implementations (like R’s defaults) are somewhat dated

  • Written in Fortran and designed for a single cpu core

  • Certain (potentially non-optimal) hard coded defaults (e.g. block size).

Multithreaded alternatives:

  • ATLAS - Automatically Tuned Linear Algebra Software

  • OpenBLAS - fork of GotoBLAS from TACC at UTexas

  • Intel MKL - Math Kernel Library, part of Intel’s commercial compiler tools

  • cuBLAS / Magma - GPU libraries from Nvidia and UTK respectively

  • Accelerate / vecLib - Apple’s framework for GPU and multicore computing

OpenBLAS Matrix Multiply Performance

x=matrix(runif(5000^2),ncol=5000)

sizes = c(100,500,1000,2000,3000,4000,5000)
cores = c(1,2,4,8,16)

sapply(
  cores, 
  function(n_cores) 
  {
    flexiblas::flexiblas_set_num_threads(n_cores)
    sapply(
      sizes, 
      function(s) 
      {
        y = x[1:s,1:s]
        system.time(y %*% y)[3]
      }
    )
  }
)

n 1 core 2 cores 4 cores 8 cores 16 cores
100 0.000 0.000 0.000 0.000 0.000
500 0.004 0.003 0.002 0.002 0.004
1000 0.028 0.016 0.010 0.007 0.009
2000 0.207 0.110 0.058 0.035 0.039
3000 0.679 0.352 0.183 0.103 0.081
4000 1.587 0.816 0.418 0.227 0.145
5000 3.104 1.583 0.807 0.453 0.266